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• Level: GCSE
• Subject: Maths
• Word count: 5692

# Statistics &quot;Mayfield high school&quot; - the taller the person the more they weigh.

Extracts from this document...

Introduction

Raheem Mirza                Mathematic Coursework

Candidate No : 2149

MATHS COURSEWORK

STATISTICS

“MAYFIELD HIGH SCHOOL”

STUDENT: RAHEEM MIRZA

TEACHER: MS LEWIS

## Part 1: Introduction/Aim

I imagine that the taller the person is the more they weigh, due to body mass. I also assume that the gender doesn’t have an effect on their height and weight. I will try and prove or disprove this in my investigation. I will take people from Key Stage 3.

Part 2: Method:

## Part 3: My 40 Random Result

 Random Result Number Year Group Gender Height(cm) Weight(kg) 671 1 9 Female 159 50 687 2 9 Female 156 50 261 3 7 Male 156 39 342 4 8 Female 175 64 584 5 9 Female 180 82 445 6 8 Male 167 51 388 7 8 Female 180 58 428 8 8 Male 154 37 646 9 9 Female 165 52 291 10 8 Female 168 59 224 11 7 Male 148 26 309 12 8 Female 163 42 654 13 9 Female 165 49 469 14 8 Male 165 40 262 15 7 Male 149 35 353 16 8 Female 160 46 635 17 9 Female 160 40 804 18 9 Male 180 42 685 19 9 Female 152 50 388 20 8 Female 180 58 479 21 8 Male 168 51 75 22 7 Female 166 45 94 23 7 Female 173 51 552 24 9 Female 160 60 548 25 8 Male 152 45 468 26 8 Male 182 54 238 27 7 Male 160 40 630 28 9 Female 162 41 750 29 9 Male 180 55 258 30 7 Male 157 48 271 31 7 Male 151 39 560 32 9 Female 152 46 739 33 9 Male 160 68 792 34 9 Male 170 50 416 35 8 Male 148 40 684 36 9 Female 158 55 335 37 8 Female 175 55 430 38 8 Male 162 53 176 39 7 Male 152 32 470 40 8 Male 165 59

Types of Formulae

I will use the following types of formulae;

• Quartile Ranges
• Mean, Mode, Median and Range
• Pearson’s Product Moment Correlation Co-Efficient

To represent my data, I intend to use the following types of Graphs;

• Cumulative Frequency Graphs
• Histograms
• Frequency Polygons
• Box and Whisker Plots

Part 5:

I will now work out the mean, mode, range and median of the data. I have decided to use intervals because height and weight are continuous data.

Frequency table for height of boy’s and girls

 Height Interval Tally Frequency (f) Midpoint (x) f (x) 148 – 154 9 151 1359 155 – 161 10 158 1580 162 – 168 11 135 1485 169 – 175 4 172 688 176 – 182 6 179 1074 Total: ∑f=40 ∑x=795 ∑fx=6188

Mean: 6188/40 = 154.7

Mode Interval: 162 – 168

Median: 162 – 168

Range: 182 – 148 =34

Frequency table for weights of boy’s and girls

 Weight Interval Tally Frequency (f) Midpoint (x) f (x) 26 –32 2 29 58 33 – 39 4 36 144 40 – 46 11 43 473 47 – 53 11 50 550 54 – 60 8 57 456 61 – 67 2 64 128 68 – 74 1 71 71 75 – 81 0 78 0 82 – 88 1 85 85 Total: ∑f=40 ∑x=513 ∑fx=1965

Mean: 1965/40 =49.13    2dp

Modal Interval: 40 – 46 and 47 – 53

Median: 47 – 53

Range: 82 – 26 =56

## Part 6: PMCC for my 40 students

 Random Result Number Height  (y)(cm) Weight  (x)(kg) xy 671 1 159 50 7950 687 2 156 50 7800 261 3 156 39 6084 342 4 175 64 11200 584 5 180 82 10440 445 6 167 51 5698 388 7 180 58 8580 428 8 154 37 9912 646 9 165 52 3848 291 10 168 59 6846 224 11 148 26 8085 309 12 163 42 6600 654 13 165 49 5215 469 14 165 40 7300 262 15 149 35 6400 353 16 160 46 7560 635 17 160 40 7600 804 18 180 42 10440 685 19 152 50 8568 388 20 180 58 7470 479 21 168 51 8823 75 22 166 45 7470 94 23 173 51 8823 552 24 160 60 9600 548 25 152 45 6840 468 26 182 54 11648 238 27 160 40 6400 630 28 162 41 6642 750 29 180 55 9900 258 30 157 48 7536 271 31 151 39 5889 560 32 152 46 6992 739 33 160 68 10880 792 34 170 50 8500 416 35 148 40 5920 684 36 158 55 8690 335 37 175 55 9625 430 38 162 53 8586 176 39 152 32 4864 470 40 165 59 9735 n=40 ∑y=1930 ∑x=6527 ∑xy=318312

x² =1068757      y² =96454

Formulae for PMCC

The information for the following formulae is above, where the PMCC for the students is worked out.

Middle

Tally

Frequency (f)

Midpoint (x)

F (x)

152 – 158

4

155

620

159 – 165

8

162

1296

166 – 172

2

169

338

173 – 179

3

176

528

180 - 186

3

183

549

Total:

f=20

x=845

fx=3331

Mean:  3331/20 = 166.55

Mode: 159 – 165

Median: 159 – 165

Range: 180-152 =28

Group frequency table for the height of boy’s

 Height Tally Frequency (f) Midpoint (x) F (x) 148 – 154 7 151 1057 155 – 161 4 158 632 162 – 168 5 165 825 169 – 175 1 172 172 176 – 182 3 179 537 Total: ∑f=20 ∑x=825 ∑fx=3223

Mean: 3223/20 = 161.15

Mode: 148 – 154

Median: 155 – 161

Range: 182 – 148 =34

Group frequency table for weight of girl’s

 Weight (kg) Tally Frequency (f) Midpoint (x) F (x) 40 – 46 6 43 258 47 – 53 6 50 300 54 – 60 6 57 342 61 – 67 1 64 64 68 – 74 0 71 0 75 – 81 0 78 0 82 - 88 1 85 85 Total: ∑f=20 ∑x=448 ∑fx=1049

Mean:  1049/20 =52045

Mode:  40 – 46, 47 – 53 and 54 – 60

Median:  47 – 53

Range: 82 –40 =42

Group frequency table for weight of boy’s

 Weight (kg) Tally Frequency (f) Midpoint (x) F (x) 26 – 32 2 29 58 33 – 39 4 36 160 40 – 46 5 43 215 47 – 53 5 50 250 54 – 60 2 57 114 61 – 67 1 64 64 68 - 74 1 71 71 Total: ∑f=20 ∑x=350 ∑fx=932

Mean: 932/20 = 46.6

Mode: 40 – 46 and 47 – 53

Median: 40 – 46

Range: 68 – 26 =42

### Part 9: PMCC for height and weight for females

 Number Height (y) (cm) Weight (x) (kg) xy 1 159 50 7950 2 156 50 7800 3 175 64 11200 4 180 82 14760 5 180 58 10440 6 165 52 8580 7 168 59 9912 8 163 42 6846 9 165 49 8085 10 160 46 7360 11 160 40 6400 12 152 50 7600 13 180 58 10440 14 166 45 7470 15 173 51 8823 16 160 60 9600 17 162 41 6642 18 152 46 6992 19 158 55 5690 20 175 55 9625 n=20 ∑y=1053 ∑x=3309 ∑xy=14818

x²=549011            y²=57187

Sxx =549011 – 3309 ²/20 =1536.95

Syy =57187 – 1053²/20 =1746.55

Sxy =175215 – 3309*1053/20 =996.15

r =996.15/1746.55*1536.95 =996.15/1638.401667 =0.6

### Part 10: PMCC for male height and weight

 Number Height (y) (cm) Weight (x) (kg) xy 1 156 39 6084 2 167 51 8517 3 154 37 5698 4 148 26 3848 5 165 40 6600 6 149 35 5215 7 180 42 7560 8 168 51 8568 9 152 45 6840 10 182 64 11648 11 160 40 6400 12 180 55 9900 13 157 48 7536 14 151 39 5889 15 160 68 10880 16 170 50 8500 17 148 40 5920 18 162 53 8586 19 152 32 4864 20 165 59 9735 n =20 ∑y =3226 ∑x =914 ∑xy =4140

x²=522550                     y²=43966

Sxx =522550 – 3226²/20 =2196.2

Syy =43966 – 914²/20 =2196.2

Sxy =148818 – 3226*914/2196.2 =1389.8

r = 1389.8/2196.2*2196.2 =1389.8/2196.2 =0.6

Part 13:

Now I will find the equation of the line of best fit in all three scatter graphs.

Equation of the line of scatter graph 1 considering the point’s A and B.

All of my samples

A =(65,178)

B =(33.5,150)

Gradient =(y2 – y1)/(x2  - x1)

=178-150/65-33.5

=8/9

y =8/9x + c

Intercept (c) using co-ordinate (65,178)

178 =(8/9*65) + c

178 =(57078) + c

c =120.22                     2dp

Equation of line of scatter graph 1

y =8/9x + 120.22

h =8/9w + 120.22

Were h =height and w =weight

Equation of line scatter graph 2 considering point’s A and B

Boys Sample

A =(55,178)

B =(34.5,140)

Gradient =(y2 – y1)/(x2  - x1)

=178 – 140/55 – 34.5

=76/41

y =76/41x + c

Intercept (c) using co-ordinate (55,178)

178 =(76/41*55) + c                                     2dp

178 – 101.95 + c

c =76.05                                                      2dp

Equation of the line of scatter graph 2

y =76/41x + 76.05

h =76/41w + 76.05

Were h =height and w =weight

Equation of the line of scatter graph 3 considering point’s A and B

Girls Sample

A =(80,180)

B =(40.5,160)

Gradient =(y2 – y1)/(x2  - x1)

=180 – 160/80 – 40.5

=40/79

y =40/79x + c

Intercept (c) using co-ordinate (80,180)

180 =(40/79*80) + c

180 – 40.51 = c

c =139.49                                                     2dp

Equation of the line of scatter graph 3

y =40/79x + 139.49

h =40/79w + 139.49

Were h =height and w =weight

Now I will explain how the equation of the best-fit line can be used. If the weight is known of a student then the height can be estimated using the equation. Also if the height is known the weight can be estimated.

Using my found Formulae(s)

## Example 1

Using the line of scatter graph 3 (girls)

h =40/79w + 139.49

The weight of a female student is 82 kg. The height can be estimated using the equation.

h =(40/79) + 139.49

h =41.51 + 139.49

h =181.01                         2dp

## Example 2

The height of a female student is 180 cm. The weight can be estimated using the equation.

h =40/79w + 139.49

180 =40/79w + 139.49

180 – 139.49 =40.51

w =40.51*70/40

w =70.89                          2dp

This suggests that if there was a male student with a height of 180 cm then his weight is estimated to be 70.893 3dp

### Table A

Cumulative frequency table for the height of girls

 Height Frequency Cumulative Frequency 152 – 158 4 4 159 – 165 8 12 166 – 172 2 14 173 – 179 3 17 180 - 186 3 20

See Graph 1

## Table B

Cumulative frequency table for the height of boys

 Height Frequency Cumulative Frequency 148 – 154 7 7 155 – 161 4 11 162 – 168 5 16 169 – 175 1 17 176 – 182 3 20

See Graph 2

Table C

Cumulative frequency table for the weight of girls

 Weight Frequency Cumulative Frequency 40 – 46 6 6 47 – 53 6 12 54 – 60 6 18 61 – 67 1 19 68 – 74 0 19 75 – 81 0 19 82 – 88 1 20

## Table D

Cumulative frequency table for the weight of boys

 Weight Frequency Cumulative Frequency 26 – 32 2 2 33 – 39 4 6 40 – 46 5 11 47 – 53 5 16 54 – 60 2 18 61 – 67 1 19 68 – 74 1 20

Conclusion

1.52

45

77

11

Female

1.59

42

78

11

Female

1.63

38

79

11

Female

1.62

38

80

11

Female

1.70

60

81

11

Female

1.58

48

82

9

Male

1.60

60

83

9

Male

1.56

60

84

9

Male

1.66

54

85

9

Male

1.66

70

86

9

Male

1.52

52

87

9

Male

1.75

75

88

9

Male

1.65

45

89

9

Male

1.52

54

90

9

Male

1.67

54

91

9

Male

1.71

60

92

9

Male

1.80

48

93

9

Male

1.73

66

94

9

Male

1.55

50

95

9

Male

1.77

66

96

9

Male

1.73

52

97

9

Male

1.54

44

98

9

Male

1.47

42

99

9

Male

1.75

63

100

9

Male

1.62

40

101

9

Male

1.46

45

102

9

Male

1.80

64

103

9

Male

1.50

70

104

9

Male

1.61

38

105

9

Male

1.54

60

106

9

Male

1.55

51

107

7

Male

1.49

47

108

7

Male

1.48

47

109

7

Male

1.63

60

110

7

Male

1.50

40

111

7

Male

1.53

35

112

7

Male

1.54

48

113

7

Male

1.54

51.5

114

7

Male

1.48

26

115

7

Male

1.61

56

116

7

Male

1.65

41

117

7

Male

1.30

35

118

7

Male

1.61

56

119

7

Male

1.47

47

120

7

Male

1.67

60

121

7

Male

1.61

46

122

7

Male

1.50

51

123

7

Male

1.47

45

124

7

Male

1.67

53

125

7

Male

1.50

56

126

7

Male

1.52

45

127

7

Male

1.63

60

128

7

Male

1.43

41

129

7

Male

1.45

31

130

7

Male

1.58

48

131

7

Male

1.60

40

132

7

Male

1.65

35

133

7

Male

1.63

50

134

11

Male

1.94

80

135

11

Male

1.69

65

136

11

Male

1.67

60

137

11

Male

1.5

35

138

11

Male

1.86

80

139

11

Male

1.68

63

140

11

Male

1.68

63

141

11

Male

1.83

75

142

11

Male

1.68

56

143

11

Male

1.65

47

144

11

Male

1.62

92

145

11

Male

1.8

68

146

11

Male

1.68

58

147

11

Male

1.71

54

148

11

Male

1.7

56

149

11

Male

1.62

50

150

11

Male

1.71

57

From the graph we can see there is a strong positive correlation. However to use a statistical way to prove this I will use the Pearson’s Product Moment Correlation Co-Efficient. I have discovered a key in excel which automatically does this calculation without making conscious though.

Below is a guide to use of this control;

This then brings up a screen, I then pressed on ‘Pearson’s’, this automatically brings up the PMCC of the sample.

Product Moment Correlation Co-Efficient of height and weight of boys and girls was calculated to be 0.568279. This means it is a “Moderate Correlation”.

Now I will create a scatter graph of male students which are inclusive in my sample. This can be viewed below;

Similarly, as for this graph from the naked eye the graph seems to be positively correlated. But using Pearson’s Product Moment Correlation Co-Efficient, we will see how much it is correlated, the formulae accessible to us tells us the PMCC between height and weight of boys is 0.639816. This again shows us it is a‘Moderate Correlation’.

Now I will create a scatter graph of female students which are inclusive in my sample. This can be viewed below;

This graph doesn’t really have a correlation to it but using PMCC the correlation is 0.483991. This again is a Moderate correlation.

Extension conclusion

In my initial conclusion, I stated girls were generally heavier than boys. The conclusion which I have come to now is the same as my initial hypothesis. Boys are generally heavier than girls.

-  -

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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